- moja ~ 1
- mbili ~ 2
- tatu ~ 3
- nne ~ 4
- tano ~ 5
- sita ~ 6
- saba ~ 7
- nane ~ 8
- tisa ~ 9
- kumi ~ 10
If we want to say eleven in Kiswahili we’d say kumi na moja which is literally 10 + 1. Twelve becomes kumi na mbili (10 + 2), and thirteen is kumi na tatu (10 + 3). Sixty is sitini (6 x 10) and seventy is sabini (7 x 10). The pattern is only broken where we find Arabic influence enter into the language.
The number system that the English-speaking world uses is also a decimal (base 10) number system. But not all number systems in the world are base 10. The Yasayama of the DRC have a base 5 number system. Their number system is based on the numbers:
- omoko ~ 1
- bafe ~ 2
- basasu ~ 3
- bane ~ 4
- lioke ~ 5
So in the Yasayama language, the word for six is lioke lomoko (5 + 1), the word for seven is lioke lafe (5 + 2), the word for eight is lioke lasasu (5 + 3), and so on.
The Yorùbá of Nigeria and Benin have a vegesimal (base 20) number system. The Khoisan of southern Africa have a binary (base 2) number system. In fact, on the continent of Africa, base 5, base 10, base 20, and base 2 are the most common number systems. But why are these number systems so common?
The base 5 and base 10 number systems likely evolved from our anatomy. Humans have 10 fingers – five on each hand – and thus base 5 and base 10. If we include our total number of fingers and toes then base 20 also makes sense. But what about base 2? Why was this a common number system in antiquity?
The 20,000 year old Ishango Bone – one of the world’s oldest mathematical artifacts – has been found to contain markings indicating an early binary number system. The classical mathematics of Nile Valley civilizations would later build upon this binary foundation. If other number systems that developed in Africa were related to our human anatomy, then could this also be true for the binary number system? The answer is a resounding ndio!
The binary number system is a system of doubling. It is the ones and zeroes of binary code that underlies modern computing. Think of the place values of our decimal number system: the one’s place, the ten’s place, the hundred’s place, thousands, ten-thousands, and so on. Where do these numbers come from?
1, 10, 100, 1000, 10000… are all powers of 10. That is:
100 = 1 (one’s place), 101 = 10 (ten’s place), 102 = 100 (hundred’s place), 103 = 1,000, 104 = 10,000
The 10 is the base of this number system. The binary system works the same way, except that the base is 2. So what are the place values in the binary number system?
- 20 = 1
- 21 = 2
- 22 = 4
- 23 = 8
- 24 = 16
- 25 = 32
These are all powers of 2! Therefore the “place values” of the binary number system are the one’s place, the two’s place, the four’s place, the eight’s place, and so on. We see that the place values are doubling: 1, 2, 4, 8, 16, 32, 64… A binary number system simply means doubling.
With that, can you guess what human biological function this relates to? Humans “double” by giving birth! The binary number system was born from the simple observation of our family ancestor tree.
There is 1 of us. Each of us has 2 parents, 4 grandparents, 8 great-grandparents, 16 great-great grandparents, 32, 64, 128, 256…We are all binary representatives of our family ancestor tree. The sequence 1, 2, 4, 8, 16… is known in mathematics as a geometric sequence with a common ratio of r = 2. This means that for each progressive generation we have twice the ancestors as the previous generation. How would you calculate the number of ancestors that you had in the generation 25 generations before the present? In Black Radical Algebra II we learn how to answer this question by finding the nth term of a geometric sequence using exponential equations.
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In African philosophical understanding, numbers are understood to be symbolic and metaphorical.
Thus number is essential in the realization of the universe. Everything in the universe is assigned a number (or number of signs). In this way Amma used the principle of number to organize the creation. Indeed, the Dogon believe that the po will contain the entire universe conceived in Amma’s thought through the concept of number.
~Dr. Chukwunyere Kamalu, Word at Face Value
One number that is important in African cosmology is the number four, or as we call it, the fantastic 4. On the most basic level, the number four represents the the four elements (earth, wind, fire, water) and the four cardinal directions (north, south, east, west).
The fantastic 4 also relates to many traditional accounts of creation that we hear across the Continent that all seem to be attempting to tell some version of the same story… That is, in many traditional creation stories the original ancestors of human beings are described as beginning from four (or sometimes four pairs) of male/female twins.
We also see the fantastic 4 represented as four quadrants in the symbology of the ancient African world.
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The fantastic 4 can be observed in traditional divination systems. Mẹ́rìndínlógún (literally “sixteen”) cowrie shell divination – used by Olórìṣà in the Yorùbá traditional religion – uses 16 cowrie shells: 16 = 42. Diviners of Ifá in the Yorùbá traditional religion use the ọ̀pẹ̀lẹ̀ divining chain or the ikin Ifá (sacred Ifá palm nuts) and the ọpọ́n Ifá (divining tray) to mark out one of 256 possible Odù Ifá signs. Each sign contains a number of verses and collectively they represent the ẹsẹ Ifá corpus – a collection of oral narratives that represent all possible scenarios and possibilities of human existence and interaction as well as the prescription to overcome any of life’s challenges. There are 256 odù signs: 256 = 44.
The Dogon of Mali have an alphabet and a divination system that is comprised of 266 signs. From Dr. Kamalu’s quote above, we know that in Dogon cosmology, Amma (the supreme being) organized the universe according to the concept of number. That is, Amma assigned everything in the universe one or more of the 266 signs mentioned earlier. For example, in Dogon cosmology the male number is 3 and the female number is 4. Together, their sum (7) represents life. The number 7 tends to hold special significance in traditional African religions.
The 266 signs that comprise the universe in Dogon cosmology are non-coincidentally similar to the 266 days between the conception and birth of a newborn child. Beyond that, the 266 signs of the Dogon actually break down further. There are two guide signs and eight master signs which serve as the keys to understanding the other 256 signs. Therefore it is: 2 + 8 + 256 = 266. This is not dissimilar to the 16 principle ojú odù of the Ifá (Yorùbá) system which are paired together to give birth to all of the 256 odù Ifá signs (256 = 162).
The Bamana of Mali use a system of divination that is also similar to the Ifá system. Bamana specialists perform their divination in the sand in a process known as cenda. The process of cenda divination begins with the specialist drawing four dashed lines horizontally in the sand.
The dashes are then paired together. If a row has an odd number of dashes then the result of that row is a single vertical mark. If the row has an even number of dashes then the result is a double vertical mark.
This is repeated four times in a recursive process that results in a total of 16 symbols (16 = 42) by the end. The 15th symbol is known as “this world” and the 16th symbol “the next world.” From there, the specialist dictates a narrative (reminiscent of the ẹsẹ Ifá oral narratives) from the symbols that addresses the original reason for the divination.
This is Afrikan math.
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Spirals are found throughout nature. In geometry, the sacred (golden) spiral is a logarithmic spiral with a growth factor equal to 1.61803…the golden ratio! We find this ratio embedded in African architecture from Cameroon to the ancient Nile Valley civilizations going back as far as 2580 BCE.
Let’s start with the golden ratio and work our way back to the golden spiral. The sacred ratio occurs when the ratio of two numbers is equivalent to the ratio of their sum to the larger quantity. Mathematically it looks like this:
When this occurs, the ratio is always the same:
The ratio is conventionally represented by the Greek letter φ (phi), but since the Greek alphabet had not yet been invented when this ratio was first being embedded into ancient African architecture, we choose to break with that convention.
Ipet-Isut (the most select/holy of places) is an ancient Nile Valley temple that housed shrines to Amun-Ra, Mut, and Montu. Known today as the Karnak Temple Complex, it sits on the east bank of the Nile River facing west in such a way that the central corridor when it was first built in 1700 BCE was directly aligned to the western hills such that on June 21st (the summer solstice) when the sun set behind the western hills, just as it was setting would send a flash of light directly down the corridor and it would shine on a metal apparatus in the sanctuary for 2 – 5 minutes. This was how these ancient Africans knew that it was the first day of summer. In this way, the central corridor of Ipet-Isut was designed with such precision that it functioned both as an architectural masterpiece and as a solar calendar.
These images show the size and scale of the Ipet-Isut Temple Complex and it’s interior – we can see that it is essentially a small town. Ipet-Isut is located in the ancient city of Waset (Thebes) – the land of the scepter.
Something else that’s special about Ipet-Isut is that the sacred ratio is embedded into it’s architecture.
If we look at a plan view of the site we’ll see numerous sacred (golden) rectangles that form a self-repeating pattern:
Sacred rectangles have dimensions that are of the sacred ratio.
If you research the golden ratio in ancient architecture you’re likely to read about the Greek Parthenon as the earliest example of the golden ratio being embedded into ancient architecture. But that is inaccurate. While the Parthenon was constructed in 490 BCE, keep in mind that the Ipet-Isut Temple Complex was constructed around 1700 BCE. As in all things mathematical, the numbers don’t lie! Remember what is first is sacred, and being the first humans on our planet, it stands to reason that Africans would be the first to advance mathematics and the sciences.
Now let’s travel over to West Africa. The photo below shows an aerial photo of the Palace of the Chief in Logone-Birni, Cameroon which is built using fractal geometry. The palace also incorporates the sacred rectangle – see if you can spot the pattern.
Logone-Birni (Fort Logone) is the capital of the Kotoko people and was founded around 1700 ACE. At the center of the palace is a sacred shrine. To get to the shrine you have to travel through a passage which is a rectangular spiral. As you get closer to the center, the size of the compound shrinks accordingly. And as you enter each progressively smaller room you are required to behave more politely. By the time you have arrived at the throne you must behave with impeccable manners and cultured formality.
…
In addition to sacred rectangles we also have sacred triangles. The Great Pyramid of Giza (2580 BCE) is the earliest example of the sacred triangle being embedded in architecture. The images below show some of the basics, but we’ll save the breakdown for another post.
Now, let’s break down the golden spiral. Earlier we said that the golden spiral is a logarithmic spiral with a growth factor equal to the golden ratio (1.61803…). So first, what is a logarithm? A logarithm is a function that is used to undo an exponent. Similar to how we use division to undo multiplication and we use subtraction to undo addition, we can use logarithms to solve for the variable when it is an exponent. For example, if you were given the equation 3x = 19,683 you can use a logarithm to solve for x. One of the properties of logarithms states that log(ax) = x·log(a) so we can transform the equation in the following manner and use the log function on our calculator to solve:
3x = 19,683
log(3x)= log(19,683)
x·log(3) = log(19,683)
x = log(19,683) ÷ log(3)
x = 9
So, back to spirals. Spirals can be found in shells, the cochlea of the ear, and the movement of whirlpools and hurricanes. Spirals are one way nature packs a lot of space into a small area. Ancient mathematicians and engineers chose to use what works in nature for their human designs. The field of permaculture encompasses these ideas today. For example, we can use natural patterns – such as spirals – in designing a compact garden.
The spiral is also an important symbol in African cosmology. Take a look at these excerpts from both the Big Bang Hypothesis and the Dogon’s description of the origin of the Universe.
| Big Bang Hypothesis (1920s ACE) | The Dogon (3200 BCE) |
|---|---|
| About 15 billion years ago, the entire universe was compacted into a single point called a singularity. The sphere/singularity exploded in all directions, and a giant cloud was formed. Some parts of the cloud moved faster than others, and many parts condensed into galaxies. Billions of these galaxies were formed. Within these galaxies, dust and gases spiraled around, collecting other pieces of gas and dust. Gravity pulled the gas and dust together into a flat disk as it spiraled around. These eventually became stars and planets. | When Amma (the supreme being) began the new creation Amma formed the germs of the new seeds, which would be used to make the second world. The first and most important, but smallest, of all seeds was the Po. The spiral at the center of the first yala (thought image) represents the movement of the po seed. But the spiraling po, which moves inside of Amma’s egg/womb reversed its direction and exploded out of Amma’s womb. This is symbolic of the “opening of Amma’s eyes” and the bursting forth from the hole of a light that lightened up the universe and revealed the existence of all things. These eyes also created the potential for the coming into being of space. The spiral of the po represents the spiral shape of the milky way galaxy, which includes the entire stellar world of which our earth is a part. |
Finally, sacred spirals are self-similar. That is, they form an infinite pattern as the spiral is magnified.
With these secrets that we’ve just shared with you hopefully you understand better why the ancient mathematical textbook – the Iahmesu Mathematical Papyrus – was titled: Correct method of investigating nature to know all that exists, all mysteries and all things secret. This knowledge was sacred to our Ancestors. How will you use it today?
]]>A junk rep is a repetition of an exercise that has little training effect. That is, because of poor form or execution, it does not significantly contribute to muscle growth or strength gains. Muhammad Ali famously noted that:
I don’t count my sit-ups. I only start counting when it starts hurting…
The idea here is that when the reps start hurting is when he knows that they are effective reps. For a good example of a junk rep versus an effective rep watch a video of a crossfitter performing kipping or butterfly pullups and contrast it with an athlete performing strict pullups. The kipping pullup is minimally effective for strength gains, therefore, compared to the strict pullup we can call them junk reps. Junk reps are inefficient at making you strong.
Homework and study are very much analogous to strength training. Successful study requires intensity (appropriately difficult problems), volume (number of homework problems worked), frequency (how regularly you’re working homework problems), and recovery (adequate sleep and nutrition). Although many students consistently complete their homework assignments, often times their work is mostly junk reps!
You get your assignment and immediately dive into it with the goal of completing the assignment as fast as possible. You’re not focused and you do not commit a conscious effort to understanding what you’re doing. You’re on auto-pilot. The result is a completed homework assignment but a sub-optimal quiz score (training effect).
If you want to become intellectually ripped, swole, and jacked then you must focus on effective reps when completing your assignments. Follow these tips:
- Ritualize your study time.
- Digest the assignment before beginning. Do this by surveying the problems and taking note of the different types of problems that the assignment contains. For example, an assignment on graphing linear equations might contain problems focusing on plotting points on the coordinate grid, identifying the slope and y-intercept from an equation in slope-intercept form, graphing a line from an equation in slope-intercept form, and graphing a line from an equation in standard form. These are the main skills that the assignment is helping you to master. Make note of this.
- Focus on accuracy and not speed. Speed comes from familiarity, and familiarity comes from volume and frequency. Do not try to rush through problems, instead sit with each problem and identify the potential challenge or lesson in the problem and then solve the problem accurately.
- Check your answers once you’ve completed the assignment. If the assignment is not online and does not offer immediate feedback then get with a classmate and check your answers against theirs. Where you have differing answers go back and check to see if you can find a mistake.
- Rework any missed problems and try to find your mistake. If after reworking the problem a couple of times you are still unable to arrive at the correct answer then ask your teacher or tutor for help.
Now get to work!
]]>One fundamental tenant of the African worldview is the idea that what is first is sacred and by imitating what is sacred, we too become sacred. Given these ideas, we asked students to discuss the connection between our topic and the Wọlọf proverb below. The top student responses follow. Wohu no sɛn? How do you see it?
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Ganaawu yaay; garabu liir la.
Mother’s back is the baby’s medicine.
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I think this proverb means that anything that helps us grow is like medicine. For example doing math is like medicine because it makes our brain stronger. Exercising is like medicine because it helps strengthen our immune systems. Medicine is not only something given to us by a doctor. Medicine can be anything that helps us get stronger.
Doing math is like taking medicine. Math is like medicine because it helps us become smarter. We use math to solve problems. Math can help us think clearly. It is like medicine for our mind. Being smart is important because it helps us to make better decisions as we grow.
Exercising is like medicine because it helps strengthen our immune system. When our immune system is strong, we are less likely to get sick. If we do get sick, a strong immune system can help us heal faster. Medicine can also help us to feel better when we are sick . Exercise and medicine can work the same way.
Medicine is anything that helps us grow. A mother is a baby’s medicine because she helps the baby get stronger as he or she grows. The mother makes sure her baby has what he or she needs. A mother also carries a baby on her back to give support. Medicine can be many things.
~Niara M.
Black Radical Pre-Algebra
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]]>The literal meaning of this proverb would mean the mother’s back helps the baby. The baby uses the mother’s back to get around since the baby itself cannot walk. This relates to medicine because you need medicine when your body cannot fight something off on its own. The proverb could also mean everything has a foundation.
Everything comes from something that came before it. It’s origins hold up that idea or thing. Without an origin, the thing would not exist. For example, without looking at the origins of math we would not have the math we use today nor would we know how to do it. The first to come paved the way for the later generations. Without the first generations, there would be nothing right now.
This proverb could be applied to life when you are having trouble with a topic, or that topic does not make sense. You could go back to its origins/basics, and possibly develop a better understanding of what you’re doing. You could also use this if there is a problem and you are trying to figure out how it happened. You could go back to the basis of the issue, and use what you find to help you prevent future issues of the type.
I think this proverb is important because if you want to understand something or be good at something you need to fully understand the basics/origins of what you’re doing.
~Jabea M.
Black Radical Algebra I